My guess is that your browser also sends your internal address towards the site, or it is given by some sort of Client-side scripting.
If you use a proxy-server it will probably be covered by the proxy in this case because the proxy is browsing for itself (making the request and so on)
The package arriving at your destination is indeed covered by NAT so the 'Source IP' is your external IP.
After using a proxy on this side my internal ip was hidden.
With a quick search on google I came on the "HTTP_X_FORWARDED_FOR "
header, it seems this is the one why they can see your private ip-address